The Complete (Lp , Lp ) Mapping Properties of Some Oscillatory Integrals in Several Dimensions

We prove that the operators $\int{_{\mathbb{R}_{+}^{2}}{{e}^{i{{x}^{a}}\cdot {{y}^{b}}}\varphi (x,y)f(y)dy}$ map ${{L}^{p}}({{\mathbb{R}}^{2}})$ into itself for $p\,\in \,J\,=\,\,\left[ \frac{{{a}_{1}}+{{b}_{1}}}{{{a}_{1}}+(\frac{{{b}_{1}}r}{2})},\frac{{{a}_{1}}+{{b}_{1}}}{{{a}_{1}}+(1-\frac{r}{2})} \right]$ if ${{a}_{l}},{{b}_{l}}\ge 1$ and $\varphi (x,y)=|x-y{{|}^{-r}},0\le r\,2$ are indicated.